k, bettas in small tanks are really unaffected by tank cycles, trust me, I have a LOT of betta tanks (over 60 jars)
Bettas do, however need good water quality, and thats were a filter can help. Biological filtration will not hurt either, but your regular weekly water changes will manage a bettas bio load just fine even if you sanitize the tank every time you clean it.
So, what a filters primary goal in a betta tank is to provide some mechanical filtration - basic removal of floating waste. You don't need a lot of current for that, and bettas don't like heavy current either.
Set the filter on its lowest speed if possible. Most bettas will learn where the current is strong and where it is weaker and will swim and sleep accordingly.
If your betta is not acting stressed out (hiding, getting caught in the intake) then leave the filter running. The water will smell better and you will likely have less algea to deal with.
Enjoy!